1. Introduction
The “time optimal” control problem is one of the most important problems in the field of control theory. The simple version is that steering the initial state 
in a Hilbert space 
to hit a target set 
in minimum time, with control subject to constraints
.
In this paper, we will focus our attention on some special aspects of minimum time problems for co-operative parabolic system involving Laplace operator with control acts in the initial conditions. In order to explain the results we have in mind, it is convenient to consider the abstract form.
Let 
and 
be two real Hilbert spaces such that 
is a dense subspace of 
Identifying the dual of 
with 
we may consider 
where the embedding is dense in the following space. Let 
be a family of continuous operators associated with a bilinear forms 
defined on 
which are satisfied with Gårding’s inequality
(1)
for 
and 
Then, from [1] and [2], for 
and 
being a bounded linear operator on 
the following abstract systems:
(2)
have a unique weak solution 
such that 
We shall denote by 
the unique solution of the Equation (2) corresponding to the control u. The time optimal control problem that we shall concern reads:
(3)
where 
is a given subset of 
which is called the target set of the Problem (3). A control 
is called a time optimal control if 
and if there is a number 
such that 
and
(4)
We call the number 
as the optimal time for the time optimal control Problem (3).
Three questions (problems) arise naturally in connection with this problem:
1) Is there a control 
and 
such that
? (this is an approximate controllability problem).
2) Assume that the answer to 1) is in the affirmative and
Is there a control 
which steers 
to hit a target set 
in minimum time?
3) If 
exists, is it unique? What additional properties does it have?
Let 
be a bounded open domain with smooth boundary 
and set 
In the works [1] and [3], the existence of time optimal controls of the following controlled linear parabolic equations with distributed control 
was obtained:
(5)
where 
is a given function in 
and 
is a closed bounded set in 
The results in [3] partly overlap with results in [1] and they were shown that if the system (5) is controllable and if 
then the corresponding time optimal control problem has at least one solution and it is bangbang.
In the work [4], the authors gave a sufficient and necessary condition for the existence of time optimal control for the problem with the target set 
and certain controlled systems. These results will be stated as follows. Consider the following controlled system
(6)
where 
is a real number. Let 
be the eigenvalues of 
with the Dirichlet boundary condition and 
be the corresponding eigenfunctions, which forms an orthogonal basis of 
We take the target set 
to be the origin 
in 
and the control set 
to be the set
where 
is a positive number, namely, 
the closed ball in 
centered at 0 and of radius 
It was proved that if 
and 
then the corresponding time optimal control problem has at least one solution if and only if 
More early, in the works [5-7], the time optimal controls problem for globally controlled linear and semilinear parabolic equations was considered.
In our papers [8,9], the time optimal control problem of 
co-operative hyperbolic systems with different cases of the observation and distributed or boundary controls constraints was considered.
In [10], optimal control of infinite order hyperbolic equation with control via initial conditions was considered.
In the present paper, the above results for the time optimal control of systems governed by parabolic equations are extended to the case of 
co-operative parabolic systems as well as control via initial conditions. First, the existence and uniqueness of solutions for 
co-operative parabolic system are proved under conditions on the coefficients stated by the principal eigenvalue of the Laplace eigenvalue problem, then the time optimal control problem is formulated and the existence of a time optimal control is proved. Then the necessary and sufficient conditions which the optimal controls must satisfy are derived in terms of the adjoint. Finally, the scaler case is given.
2. n × n Co-Operative Parabolic Systems
Let 
be the usual Sobolev space of order one which consists of all 
whose distributional derivatives 
and 
with the scalar product norm
We have the following dense embedding chain [11]
where 
is the dual of 
Here and everywhere below the vectors are denoted by bold letters. For 
and
, let us define a family of continues bilinear forms
(7)
where
(8)
The bilinear form (7) can be but in the operator form:
where 
is 
matrix operator which maps 
onto 
and takes the form
.
Lemma 2.1. If 
is a regular bounded domain in 
with boundary 
and if 
is positive on 
and smooth enough ( in particular
) then the eigenvalue problem:
possesses an infinite sequence of positive eigenvalues:
Moreover 
is simple, its associate eigenfunction 
is positive, and 
is characterized by:
(9)
Proof. See [12]. ,.
Now, let
(10)
Lemma 2.2. If (8) and (10) hold then, the bilinear form (7) satisfy the Gårding inequality
Proof. In fact
By Cauchy Schwarz inequality and (9), we obtain
From (10) we have
Add 
to two sides, then we have the result. ,.
We can now apply Theorem 1.1 and Theorem 1.2 Chapter 3 in [1] (with 
and
) to obtain the following theorem:
Theorem 2.3. If (8) and (10) hold, then there exist a unique solution
satisfying the following 
system: 
(11)
Moreover 
is continuous from 
3. Minimum Time and Controllability
We denote the unique solution of (11), at time 
for each control 
by 
Occasionally, we write 
when the explicit dependence on 
is required. We can now formulate the time optimal control problem corresponding to the 
cooperative parabolic system (11):
(12)
with constraints
(13)
and 
and 
are given.
Theorem 3.1. If (8) and (10) are hold, then the system whose state is given by (11) is controllable, i.e.,
(14)
Proof. Let us first remark that by translation we may always reduce the problem of controllability to the case were the system (11) with 
We can show quit easily that (11) is approximately controllable in 
in any finite time 
if and only if, 
is dense in 
By the Hahn-Banach theorem, this will be the case if
(15)
for all 
implies that 
Let us introduce the adjoint state 
by the solution of the following system
(16)
where 
is the adjoint of 
which is defined by
The existence of a unique solution for the Problem (16) can be proved using Theorem 2.3, with an obvious change of variables.
Multiply the first equation in (16) by 
and integrate by parts from 0 to 
we obtain the following identity:
and so, if (15) holds, then
hence 
But from the backward uniqueness property, 
and hence 
,.
Now set
(17)
Then , the following result holds.
Theorem 3.2. If (8) and (10) are hold, then there exist an admissible control 
to the problem (12)-(17), which steering 
to hitting a target set 
in minimum time 
(defined by (17)). Moreover
(18)
Proof. Fixe 
we can choose 
and admissible controls 
such that
Set 
Since 
is bounded, we may verify that 
ranges in a bounded set in
.
We may then extract a subsequence, again denoted by 
such that
(19)
We deduce from the equality
that
and
But
Now from (19)
(20)
and
(21)
Combine (20) and (21) show that
(22)
and so, 
as 
is closed and convex, hence weakly closed. This shows that 
is reached in time 
by admissible control
.
For the second part of the theorem, really, from Theorem 2.3, the mapping 
from 
is continuous for each fixed 
and so 
for any 
by minimality of 
Using Theorem 2.3, it is easy to verify that the mapping
, from
, is continuous and linear. then, the set
is the image under a linear mapping of a convex set hence 
is convex. Thus we have
and 
(boundary of
) . Since 
(from (14)) so there exists a closed hyperplane separating 
and 
containing
, i.e. there is a nonzero 
such as
(23)
From the second inequality in (23), 
must support the set 
at 
i.e.
and since 
is a Hilbert space, 
must be of the form
Dividing the inequality (23) by 
gives the desired result. ,.
Now Inequality (18) can be interpreted as follows: let us introduce the adjoint state 
by the solution of the following system
(24)
As the proof of Theorem 3.1, we multiply the first equation int (24) by 
and integrate by parts from 
to 
we obtain the following identity:
hence condition (18) becomes
(25)
Using controllability condition (14), the backward uniqueness property implies 
then the optimal control is bang-bang, i.e., 
and since
is strictly convex, then the optimal control is unique. We have thus proved:
Theorem 3.3. If (8) and (10) are hold, then there exist the adjoint state
such that the optimal control 
of problem (12)-(17) is bang-bang unique and it is determined by (24), (25) together with (11) (with
).
4. Scaler Case
Here, we take the case where 
in this case, the time optimal problem therefore is
The state 
is solution of the following equations
with
The adjoint is solution of the following equations
The maximum condition is
5. Comments
We note that, in this paper, we have chosen to treat a special systems involving Laplace operator just for simplicity. Most of the results we described in this paper apply without any change on the results to more general parabolic systems involving the following second order operator:
with sufficiently smooth coefficients (in particular,
) and under the LegendreHadamard ellipticity condition
for all 
and some constants 
In this case, we replace the first eigenvalue of the Laplace operator by the first eigenvalue of the operator 
(see [12]).
In this paper, we have chosen to treat a co-operative parabolic system with Dirichlet boundary conditions. The results can be extended to the case of 
cooperative parabolic system with Neumann boundary conditions: if we take 
instead of 
we have to replace the Dirichlet boundary conditions 
on the boundary by Neumann boundary conditions 
where 
is the outward normal.
The results in this paper carry over to the fixed-time problem ([1] chapter 3).
subject to (11) [except in the trivial case where 
for some admissible control
]. This can be proven in an analogous manner, as the necessary and sufficient conditions for optimality for this problem coincide with (11), (16) and (25) (with
).